Sufficient conditions for the existence of a path‐factor which are related to odd components

In this article, we are concerned with sufficient conditions for the existence of a { P 2 , P 2 k + 1 } ‐factor. We prove that for k ≥ 3 , there existsε k > 0 such that if a graph G satisfies∑ 0 ≤ j ≤ k − 1c 2 j + 1( G − X ) ≤ ε k | X |for all X ⊆ V ( G ) , then G has a { P 2 , P 2 k + 1 } ‐factor, wherec i ( G − X )is the number of components C of G − X with | V ( C ) | = i . On the other hand, we construct infinitely many graphs G having no { P 2 , P 2 k + 1 } ‐factor such that∑ 0 ≤ j ≤ k − 1c 2 j + 1( G − X ) ≤ 32 k + 141 72 k − 78| X |for all X ⊆ V ( G ) .